Using a Cobb Douglas production function as follows: Q = 4L1/4K 1/4 where p=price of good; w=wage and r=price capital. Derive the input demand functions for; 1 Labour; L = f (w,r, p). 2 Capital; K = f (w,r, p).
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Using a Cobb Douglas production function as follows: Q = 4L1/4K 1/4
where
p=price of good;
w=wage and
r=price capital.
Derive the input demand functions for;
1 Labour; L = f (w,r, p).
2 Capital; K = f (w,r, p).
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- Craig and Javad run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's longrun production function is Q = 4KL, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The weekly cost function for the paper plant is C = 20K + 4L, where C is the total weekly cost. (a) What ratio of capital to labor minimizes Craig and Javad's total costs? (b) How much capital and labor will Craig and Javad need to rent and hire in order to produce 1,000 reams of paper each week? (c) How much will hiring these inputs cost them?Consider a competitive firm whose Cobb-Douglas production function is f (x1, x2) where x1 denotes the amount of labor and x2 denotes the amount of capital. Suppose that the amount of capital is fixed in the short run at 2 = 100. Let the hourly wage rate be wi = $20, the capital rental rate w2 = $30 and the price of the firm's product p = $100. (a) Are the returns to scale increasing, constant or decreasing? Explain you assertion. (b) What is the short-run marginal cost function? (c) What is the average variable cost function of this firm? (d) What is the short-run output of this firm? (e) Will the firm continue to operate in the long run? Explain your assertion.Q2) Consider the following production function TPL = 12L2 – 0.8L3 Determine the marginal product function(MPL) Determine the average product function (APL) Find the value of L that maximizes TPL Find the value of L that maximizes APL Find the value of L that maximizes MPL
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- Consider a firm that produces widgets according to the following Cobb-Douglas production function: Q = A * L^α * K^β where: Q is the quantity of output, L is the quantity of labor, K is the quantity of capital, A is a scale parameter (total factor productivity), α and β are the output elasticities of labor and capital respectively. Given that A = 1, α = 0.6, β = 0.4, L = 16 and K = 9, a) Calculate the quantity of output Q. b) If the firm increases the quantity of labor (L) to 20 while keeping the quantity of capital (K) constant, what will be the new quantity of output?Assume a firm has the following short-run production function: q(L) = L+6L2-1/2L3 If this firm is going to maximize profits in the short-run by producing a strictly positive level of output (q>0), what is the lowest level of labor (L) they would employ? A.) Minimum L=?Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.
- The short run production function of a competitive firm is given The short-run production function of a competitive firm is given by f (L) = 6L2/3, where L is the amount of labor it uses. (For those who do not know calculus—if total output is aLb, where a and b are constants, and where L is the amount of some factor of production, then the marginal product of L is given by the formula abLb−1.) The cost per unit of labor is w = 6 and the price per unit of output is p = 3. (a) Plot a few points on the graph of this firm’s production function and sketch the graph of the production function, using blue ink. Use black ink to draw the isoprofit line that passes through the point (0, 12), the isoprofit line that passes through (0, 8), and the isoprofit line that passes through the point (0, 4). What is the slope of each of the isoprofit lines? (b) How many units of labor will the firm hire? (c) Suppose that the wage of labor falls to 4, and the price of output…The Cobb-Douglas production function with output Q and capital and labor inputs K and L, respectively, is given by: Q = f(K,L) = K«LB where 0 < a <1 and 0Given the production function y = ( ip - Kp)1/p , what is the technical rate of substitution, the elasticity of substitution, and the returns to scale when p = 0.5?SEE MORE QUESTIONS
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